Geodesic
Classes of piecewise quasiaffine transformations on dihedral, quasidihedral and modular maximal-cyclic 2-groups
Diskretnaya Matematika, Tome 34 (2022) no. 2, pp. 50-66

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Nonabelian 2-groups $H$ containing a cyclic subgroup of index 2 are dihedral groups, generalized quaternion groups, quasidihedral groups and modular maximal-cyclic groups. Earlier the authors introduced the classes of piecewise quasiaffine transformations on an arbitrary nonabelian 2-group $H$ with a cyclic subgroup of index 2. For the generalized group of quaternions of order $2^m$ we have obtained a complete classification of orthomorphisms, complete transformations and their left analogues in the class of piecewise quasiaffine transformations under consideration. This paper presents a similar classification for the remaining three groups (the dihedral group, the quasidihedral group and the modular maximal-cyclic group).
Keywords: orthomorphism, complete transformation, dihedral group, quasidihedral group, modular maximal-cyclic group 
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B. A. Pogorelov; M. A. Pudovkina. Classes of piecewise quasiaffine transformations on dihedral, quasidihedral and modular maximal-cyclic 2-groups. Diskretnaya Matematika, Tome 34 (2022) no. 2, pp. 50-66. http://geodesic.mathdoc.fr/item/DM_2022_34_2_a5/